Close to the \(\Gamma \) -point the surface state dispersion relation is described well by linear \(\mathbf {k}\cdot \mathbf {p}\)  theory and taken to be linear, and so the energy levels of the TQD surface states can be considered equally spaced. When higher order terms are to be taken into account, we return to the continuum model and use the dispersion relation

Ec,v = A |k| + A1 |k|2,

where \(E_\mathrm {c,v}\)  give the energies of the conduction and valence bands respectively. For Bi\(_2\) Se\(_3\) , \(A=3.0\)  eVÅ, \(A_1 = 22.2\)  eVÅ\(^2\) , and for Bi\(_2\) Te\(_3\) , \(A=2.0\)  eVÅ, \(A_1 = 35.3\)  eVÅ\(^2\)  [1]}. More details on these values are given in Supplementary Material REF . On confinement of the surface states, \(|\mathbf {k}| \sim 1/R\)  and so the correction is proportional to \(1/R^2\) , demonstrated in FIG. REF a. For a Bi\(_2\) Se\(_3\)  TQD of \(R=50\)  nm, without the correction all nearest-level transitions occur with frequency \(\nu ^{(0)} = 1.44\)  THz (asides from the transition coupling levels directly above and below the \(\Gamma \) -point, which couples at frequency \(2\nu ^{(0)}\) ), as illustrated in FIG. REF b. When taking the correction into account, transition frequencies in both the valence and conduction bands rapidly become off-resonant with the first transition in each band. The first and second transition frequencies in the upper Dirac cone show a disparity of 3\(\%\) . The first and third transition frequencies differ by 6\(\%\) , as shown in FIG. REF b. The first transition frequencies in the valence and conduction bands differ by \(\sim 9\%\) . Variation in frequency is more rapid in smaller particles, and in materials for which \(A_1\)  is large. This disparity can thus be increased by considering a TQD of smaller radius, or by using a different material such as Bi\(_2\) Te\(_3\) . It should be noted that interband transitions are unaffected due to the cancellation of the second-order correction.
